| Abstract : | It is common practice for insurance companies to give dividends to their shareholders. Many papers have been written on dividends policies. It has been found that under some reasonable assumptions the optimal policy is to follow the so called constant barrier dividends policy also known in Risk theory as de Finetti model. However soon become apparent that this model is not "perfect" as questions and problems emerge from its application and that some "kind" of modications are necessary. In this spirit, this thesis extends the de Finetti model in order to include cases with barriers dividends policies which are modeled by diffusions. The approach is axiomatic and was motivated by the classical de Finetti model. We show that the de Finetti models with general (diffusion) barriers are well posed that is they exist and are unique, or in other words that there exist unique stochastic processes that evolve according to our conditions. When we say unique stochastic processes we mean up to the degree of indistinguishability. We consider de Finetti models with one general barrier meaning that when the reserves of the insurance company reach a "particular" level, which also depends upon a diffusion process, then the company goes bankrupt. We also consider de Finetti models with two general barriers, that is when the reserves of the insurance company reach the level of the lower barrier, which also depends on a di¤usion process, then the insurance company has the option to borrow money and continue its function. We derive differential equations with appropriate boundary conditions, the solution of which gives the quantities for which we are interesting. More specifically we find differential equations with appropriate boundary conditions, the solution of which gives the moments of the discounted dividends, the moments of the discounted financing, the Laplace transform of the time of ruin, the Laplace transform of the joint distribution of the time of ruin and the discounted dividends and the Laplace transform of the joint distribution of the discounted dividends and the discounted financing. We apply the formulas in special cases and more specifically in cases where the reserves process follows a Brownian motion, a Geometric Brownian motion and an OrnsteinUhlenbeck process (also see Gerber, H.U. and Shiu, E.S.W.([71],[72])). Next we work on another important issue, which is the situation of insurance companies cooperation. We consider this issue from the perspective of a particular insurance company. We are interesting to look at parameters which are vital to the decisions of the company. Among these parameters very important role we consider to play the probability of survival in a particular cooperation and the shares that will be given to the shareholders during this cooperation. We find differential equations with appropriate boundary conditions the solution of which will give:• The moments of the discounted dividends and the discounted financing.• The Laplace transform of the joint distribution of the time of ruin and the discounted• dividends.• The Laplace transform of the discounted dividends.• The Laplace transform of the time of ruin.• The Survival probability for one of the two insurers.We apply these results in two models:(I) The Lundberg - de Finetti model.(II) The de Finetti - de Finetti model.We show how an insurance company can use the above results for policy making purposes. We also mention possible ways to extend the above considerations to various other models.
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